All Questions
26 questions
1
vote
0
answers
58
views
Can a maximal rank subgroup of a simply connected Lie group have simply connected factors?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\rank{rank}$Take a simply connected Lie group $G$ such as $\SU(N)$ and a maximal rank subgroup $H$, i.e. $\rank(G) = \rank(H)$. Assume that $H$ takes ...
77
votes
7
answers
21k
views
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
106
votes
3
answers
10k
views
Has the Lie group E8 really been detected experimentally?
A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...
- 20.7k
5
votes
3
answers
849
views
Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
- 835
4
votes
1
answer
288
views
The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators
Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$).
How is the embedding $\mathfrak{g}...
- 1,066
12
votes
2
answers
849
views
Geodesics on $SU(4)$
Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find?
In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
- 2,099
5
votes
1
answer
256
views
Definition of a Dirac operator
So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
- 327
9
votes
1
answer
444
views
Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
- 835
6
votes
0
answers
273
views
Branching rules for E6 into SU(3)^3
I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
- 489
8
votes
0
answers
381
views
Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
- 2,751
5
votes
0
answers
460
views
Chern-Simons theory with non-compact gauge groups G
This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
- 10.5k
4
votes
1
answer
923
views
About using the character formula for $SO(2n)$
I have known of the following equation for characters of a $SO(2n)$ representation with highest weights $(h_1,...,h_n)$ and for $(t_1,t_2,..,t_n,t_1^{-1},t_2^{-1},..,t_n^{-1})$ being the eigenvalues ...
- 1,893
3
votes
1
answer
1k
views
Highest weight orbit characterization (reformulated and extended)
Edit 1: I think that the question was not stated clearly enough so modified it a little.
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original ...
2
votes
1
answer
1k
views
A question about flag variety of $SL(n,\mathbb{C})$
We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of $SL(...
user21574
2
votes
0
answers
808
views
Casimir operators of a given Lie Algebra
I am a Physicist, so let me apologize in advance for some possible imprecisions.
I'm working on a 10-dimensional Lie Algebra. Each element of the algebra represents a quantum mechanical operator, and ...
- 255
10
votes
3
answers
4k
views
Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?
I am an analyst struggling through some geometry used in physics.
Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection ...
- 1,771
5
votes
0
answers
126
views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
9
votes
0
answers
360
views
Finding $U,V$ in Thompson's Formula
Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that:
$e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$
Given $a,b \in \mathfrak{su}(4)$ defined by:
$a=J_x ...
- 2,099
1
vote
2
answers
2k
views
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal ?
Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system.
My ...
- 41
2
votes
0
answers
423
views
First Variation of Dyson Series/Magnus Expansion
Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...
- 2,099
0
votes
1
answer
3k
views
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
- 41
8
votes
2
answers
1k
views
Killing form vs its counterpart in a given represenation
Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...
- 965
1
vote
0
answers
324
views
Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation
Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
- 2,099
0
votes
1
answer
313
views
Symplectic structure on $Sym^kG^{\mathbb{C}} $
Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here $G^{\mathbb{...
user21574
2
votes
1
answer
359
views
Characterization of the weight orbit in the projective space via second order Casimir.
This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
- 965
2
votes
0
answers
115
views
The condition of maximality in branching rules of $SO$ group representations
Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
- 1,893